Some arguments are worth getting into - some aren't. This isn't, believe me, I would strongly advise that you cease communication on the matter altogether because you're wasting precious time - time that could be spent honouring God in other ways.
Even if you did manage to prove inconclusively that this account was totally fictitious - it's not going to succeed in converting this person's heart, is it? They will probably come up with another account that they use to justify - to themselves - why they are against the Church, therefore you will never out-argue them. They are on a journey, but pretty blinkered at the moment as to the Truth, just pray they will find God on a personal level, because they are just seeing the Church as a physical institution at the moment.
The reason I strongly advise you to drop this - I fell into the same trap myself a few years ago, and got into an online debate with some atheists, no less, who had no intention, never had, of being swung by my arguments. Essentially they wanted to back me in a corner with intellectual and historical arguments, I wasted so much time and energy on that debate that I was exhausted and distracted about it in the rest of my life. It didn't do me any good spiritually, and my arguments didn't do them any good either, no matter how convincing they were. In the end, I got pretty emotional about it because I felt under attack from them, in a spiritual sense, and so totally withdrew from the argument. Once I took a step back from it, I realised how much time, energy and emotions I'd wasted on it, to get me... nowhere!